Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C8 · Schur multiplier (H2) = C28×C4 |
| direct product | C22×C4×M32 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C8 | C22 |
| commutator | C2 | C23×C4×C8 |
| Frattini, cosocle | C2×C8 | C25 |
| socle | C24 | C22×C8 |
| prime radical | C25 | C2×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
2794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2);(3,4);(5,6,7,8);(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24);(10,18)(12,20)(14,22)(16,24)320 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C22×C8×M32, C22×#11414, C22×#11599, … |
| maximal normal subgroup of | C22×C8×M32, C22×#11414, C22×#11599, … |
| maximal quotient of | M32×#131, C22×C8×M32, C22×#12599 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529609);